[MUSIC] So, let us consider now this killing vector 1, 0, 0, 0 which corresponds to time translations for the Schwarzschild spacetime. Then K mu times U mu = U0, = g00 U0. Remember, g00 component of the metric is just, well remembering what is the g00 component for the Schwarzchild metric. This is just 1-rg/r, times and U0, I'll remind you because, well, and this coordinate is dt/ds. So this is U0, and this is g00. So this is a conserved quantity, we will denote it as E/m, as E/m. The m is the mass of the particle. We're going to explain that E is energy, but at this moment, one can say that according to the Noether's theorem, every symmetry has as a consequence a conservation law. So the fact that this quantity is conserved is a revelation of Noether's theorem. And one knows from the classical physics that from the time translations it follows that the energy is conserved. So, this is one of the arguments that E is energy, but below we're going to see a bit more arguments why E is energy. So another killing vector that we have in Schwarzschild spacetime with upper index is just 0 ,0, 0, 1, and corresponding conserved quantity as follows from the same observations is just r squared times sin squared theta d phi / dS is conserved. So, we will denote it as L/m, where m is again mass and L is going to be angular momentum as follows from the rotational symmetry, follows the consideration of angular momentum. But we're going to see a bit more arguments, why L is momentum? So, now, to move further, let us observe that this is nothing but the Kepler's Law, basically, one of the Kepler's Laws. Because, in fact, well, a relativistic variant, because we have here dividing by proper time rather than coordinate time. But the Kepler's Law says that if we have a planet, and its radius vector during unit time, equal areas are covered by the radius vector of the planet. So, this is exactly radius vector of the planet, as one can see. And from this observation in analogy with the situation of non-relativistic situation, we can understand that the motion of the particle was mass m, goes in one plane which means that we can fix theta. And we're going to fix it to be at pi/2, so which means that the motion goes in the space defined by the equation, X3 = 0. And as a result, this conservation law has the following expression, r squared d phi / ds = L / m. So, we have these two conservation laws, this one. This one. And this one, and going to use it. Now, so remember that for a motion for the full velocity vector, we have the following equation. 1 = g mu nu, dz mu/ds, dz nu / ds. Well, just because in the denominator we have ds squared, and in the numerator we also have ds squared. That's the reason this is 1. So, this is ds squared, which means that g mu times U mu times U nu is equal to 1. Now, using this for the Schwarzchild metric. Let me remind you that the Schwarzchild metric has this form, metric tensor has this form. It's 1- rg / r- 1 / 1- rg / r and also, of course here, we have r squared. And here we have r squared sine squared theta. So this is a metric tensor, but we also take into account that this part doesn't participate, because, in our case, theta is constant. Hence, d theta here in this expression d theta = 0. So using this metric here, using the expression for the full velocity and the fact that d theta = 0, and theta = pi / 2 we have the following actuality, (1- rg / r) (dt / ds) squared- ( dr / ds ) squared / 1- rg / r- r squared d phi / ds squared. So this is equal to 1, this is equal to 1. Now, using this conservation law and this conservation law in this expression, we obtain the following equation. That (dr / ds) squared = (E / m) squared- (1- rg / r), ( 1 + L squared / m squared r squared ). So, this is an equation for the Geodesic. This is an equation that we have been looking for, but to understand the physical meaning of E and L, let us take the non relativistic limit of this equation. So, what means non relativistic limit? Non relativistic limit, first of all, is that rg is much less than r. So, we are very far from the gravitating body. And velocity dr / dt, is much less than 1, much less than the speed of light. In this limit it can be seen that dr / ds is approximately is dr / dt. Well, actually, better to say that this is much less than one. So then this is equal to. As a result, in this limit, if we take this limit in this expression, we obtain the following equation. That E squared- m squared / 2m is approximately equal m / 2 ( dr / dt ) squared + L squared / 2mr squared- mrg / 2r. So let us look more closely to this equation. Let us look closely to this equation. And this thing, in the limit that we are considering, again in non relativistic limit is nothing but E squared- m squared = ( E- m ) ( E + m ). And it is not hard to see that, in the non relativistic limit that we are considering, this is approximately 2m, and this is approximately E, which is kinetic energy. As a result, we have a product of this times this, divided by 2m. This is nothing but approximately non relativistic kinetic energy E standing here. So, this equation is nothing but non relativistic expression that we know in standard non relativistic mechanics for the motion in Newton's potential. This is because exactly expression for the Newton's potential, because I remind you that this is nothing but 2 kappa M, where kappa is Newton's constant and M is the mass of the gravitating body. And 2 is cancelled, so this is just Newton's law, this is just kinetic energy. Sorry this is the total energy rather than kinetic energy, I'm sorry.. Now, non redrustic total energy. And then this proves that L, another way to see that L, which is here, is angular momentum, and E, which stands here, is nothing but the energy. So we have everything at our disposal. Remember that this much less than m. So anyway, we have everything in our disposal to go further with a solution of this equation. [MUSIC]